On iterative methods and implicit-factorization preconditioners for regularized saddle-point systems

Abstract

We consider conjugate-gradient like methods for solving block symmetric indefinite linear systems that arise from saddle point problems or, in particular, regularizations thereof. Such methods require preconditioners that preserve certain sub-blocks from the original systems but allow considerable flexibility for the remaining blocks. We construct fourteen families of implicit factorizations that are capable of reproducing the required sub-blocks and (some) of the remainder. These generalize known implicit factorizations for the unregularized case. Improved eigenvalue clustering is possible if additionally some of the non-crucial blocks are reproduced. Numerical experiments confirm that these implicit-factorization preconditioners can be very effective in practice.